Search for: All records

Abstract We rigorously justify the mean-field limit of an N -particle system subject to Brownian motions and interacting through the Newtonian potential in $${\mathbb {R}}^3$$ R 3 . Our result leads to a derivation of the Vlasov–Poisson–Fokker–Planck (VPFP) equations from the regularized microscopic N -particle system. More precisely, we show that the maximal distance between the exact microscopic trajectories and the mean-field trajectories is bounded by $$N^{-\frac{1}{3}+\varepsilon }$$ N - 1 3 + ε ( $$\frac{1}{63}\le \varepsilon <\frac{1}{36}$$ 1 63 ≤ ε < 1 36 ) with a blob size of $$N^{-\delta }$$ N - δ ( $$\frac{1}{3}\le \delta <\frac{19}{54}-\frac{2\varepsilon }{3}$$ 1 3 ≤ δ < 19 54 - 2 ε 3 ) up to a probability of $$1-N^{-\alpha }$$ 1 - N - α for any $$\alpha >0$$ α > 0 . Moreover, we prove the convergence rate between the empirical measure associated to the regularized particle system and the solution of the VPFP equations. The technical novelty of this paper is that our estimates rely on the randomness coming from the initial data and from the Brownian motions. 

Abstract In this paper, we introduce a novel method for deriving higher order corrections to the mean-field description of the dynamics of interacting bosons. More precisely, we consider the dynamics of N $$d$$ d -dimensional bosons for large N . The bosons initially form a Bose–Einstein condensate and interact with each other via a pair potential of the form $$(N-1)^{-1}N^{d\beta }v(N^\beta \cdot )$$ ( N - 1 ) - 1 N d β v ( N β · ) for $$\beta \in [0,\frac{1}{4d})$$ β ∈ [ 0 , 1 4 d ) . We derive a sequence of N -body functions which approximate the true many-body dynamics in $$L^2({\mathbb {R}}^{dN})$$ L 2 ( R dN ) -norm to arbitrary precision in powers of $$N^{-1}$$ N - 1 . The approximating functions are constructed as Duhamel expansions of finite order in terms of the first quantised analogue of a Bogoliubov time evolution.